When linear models are fit to ill-conditioned or confounded narrow-data, TRACE plots are useful in demonstrating and justifying deliberately biased estimation. This makes TRACE diagnostics powerful "visual" displays. If advanced students of regression are trained in interpretation of Trace plots, they could help admininstrators capable of basic statistical thinking avoid misinterpretations of questionable regression coefficient estimates. All five types of ridge TRACE plots for a wide variety of ridge paths can be explored using R-functions. For example, the RXshrink aug.lars() function generates TRACE s for Least-Angle, Lasso and Forward Stagewise methods (Efron, Hastie, Johnstone and Tibshirani 2004; Hastie and

We develop a novel hybrid method for Bayesian network structure learning called partitioned hybrid greedy search (pHGS), composed of three distinct yet compatible new algorithms: Partitioned PC (pPC) accelerates skeleton learning via a divide-and-conquer strategy, $p$-value adjacency thresholding (PATH) effectively accomplishes parameter tuning with a single execution, and hybrid greedy initialization (HGI) maximally utilizes constraint-based information to obtain a high-scoring and well-performing initial graph for greedy search. We establish structure learning consistency of our algorithms in the large-sample limit, and empirically validate our methods individually and collectively through extensive numerical comparisons. The combined merits of pPC and PATH achieve significant computational reductions compared to the PC algorithm without sacrificing the accuracy of estimated structures, and our generally applicable HGI strategy reliably improves the estimation structural accuracy of popular hybrid algorithms with negligible additional computational expense. Our empirical results demonstrate the superior empirical performance of pHGS against many state-of-the-art structure learning algorithms.

In this paper we perform a numerious numerical studies for the problem of low-rank matrix completion. We compare the Bayesain approaches and a recently introduced de-biased estimator which provides a useful way to build confidence intervals of interest. From a theoretical viewpoint, the de-biased estimator comes with a sharp minimax-optinmal rate of estimation error whereas the Bayesian approach reaches this rate with an additional logarithmic factor. Our simulation studies show originally interesting results that the de-biased estimator is just as good as the Bayesain estimators. Moreover, Bayesian approaches are much more stable and can outperform the de-biased estimator in the case of small samples. However, we also find that the length of the confidence intervals revealed by the de-biased estimator for an entry is absolutely shorter than the length of the considered credible interval. These suggest further theoretical studies on the estimation error and the concentration for Bayesian methods as they are being quite limited up to present.

We present a simple algorithm for identifying and correcting real-valued noisy labels from a mixture of clean and corrupted samples using Gaussian process regression. A heteroscedastic noise model is employed, in which additive Gaussian noise terms with independent variances are associated with each and all of the observed labels. Thus, the method effectively applies a sample-specific Tikhonov regularization term, generalizing the uniform regularization prevalent in standard Gaussian process regression. Optimizing the noise model using maximum likelihood estimation leads to the containment of the GPR model's predictive error by the posterior standard deviation in leave-one-out cross-validation. A multiplicative update scheme is proposed for solving the maximum likelihood estimation problem under non-negative constraints. While we provide a proof of monotonic convergence for certain special cases, the multiplicative scheme has empirically demonstrated monotonic convergence behavior in virtually all our numerical experiments. We show that the presented method can pinpoint corrupted samples and lead to better regression models when trained on synthetic and real-world scientific data sets.

B-cell epitopes play a key role in stimulating B-cells, triggering the primary immune response which results in antibody production as well as the establishment of long-term immunity in the form of memory cells. Consequently, being able to accurately predict appropriate linear B-cell epitope regions would pave the way for the development of new protein-based vaccines. Knowing how much confidence there is in a prediction is also essential for gaining clinicians' trust in the technology. In this article, we propose a calibrated uncertainty estimation in deep learning to approximate variational Bayesian inference using MC-DropWeights to predict epitope regions using the data from the immune epitope database. Having applied this onto SARS-CoV-2, it can more reliably predict B-cell epitopes than standard methods. This will be able to identify safe and effective vaccine candidates against Covid-19.

Since the seminal work of Venkatakrishnan et al. (2013), Plug & Play (PnP) methods have become ubiquitous in Bayesian imaging. These methods derive Minimum Mean Square Error (MMSE) or Maximum A Posteriori (MAP) estimators for inverse problems in imaging by combining an explicit likelihood function with a prior that is implicitly defined by an image denoising algorithm. The PnP algorithms proposed in the literature mainly differ in the iterative schemes they use for optimisation or for sampling. In the case of optimisation schemes, some recent works guarantee the convergence to a fixed point, albeit not necessarily a MAP estimate. In the case of sampling schemes, to the best of our knowledge, there is no known proof of convergence. There also remain important open questions regarding whether the underlying Bayesian models and estimators are well defined, well-posed, and have the basic regularity properties required to support these numerical schemes. To address these limitations, this paper develops theory, methods, and provably convergent algorithms for performing Bayesian inference with PnP priors. We introduce two algorithms: 1) PnP-ULA (Unadjusted Langevin Algorithm) for Monte Carlo sampling and MMSE inference; and 2) PnP-SGD (Stochastic Gradient Descent) for MAP inference. Using recent results on the quantitative convergence of Markov chains, we establish detailed convergence guarantees for these two algorithms under realistic assumptions on the denoising operators used, with special attention to denoisers based on deep neural networks. We also show that these algorithms approximately target a decision-theoretically optimal Bayesian model that is well-posed. The proposed algorithms are demonstrated on several canonical problems such as image deblurring, inpainting, and denoising, where they are used for point estimation as well as for uncertainty visualisation and quantification.

There are numerous examples in machine learning, statistics, mathematics and deep learning, requiring an algorithm to solve some complicated equations: for instance, maximum likelihood estimation (think about logistic regression or the EM algorithm) or gradient methods (think about stochastic or swarm optimization). Here we are dealing with even more difficult problems, where the solution is not a set of optimal parameters (a finite dimensional object), but a function (an infinite dimensional object). The context is discrete, chaotic dynamical systems, with applications to weather forecasting, population growth models, complex econometric systems, image encryption, chemistry (mixtures), physics (how matter reaches an equilibrium temperature), astronomy (how celestial man-made or natural bodies end up having stable or unstable orbits), or stock market prices, to name a few. These are referred to as complex systems. The solutions to the problems discussed here requires numerical methods, as usually no exact solution is known.

Inductive inference in supervised classification context constitutes to methods and approaches to assign some objects or items into different predefined classes using a formal rule that is derived from training data and possibly some additional auxiliary information. The optimality of such an assignment varies under different conditions due to intrinsic attributes of the objects being considered for such a task. One of these cases is when all the objects' features are discrete variables with a priori known categories. As another example, one can consider a modification of this case with a priori unknown categories. These two cases are the main focus of this thesis and based on Bayesian inductive theories, de Finetti type exchangeability is a suitable assumption that facilitates the derivation of classifiers in the former scenario. On the contrary, this type of exchangeability is not applicable in the latter case, instead, it is possible to utilise the partition exchangeability due to John Kingman. These two types of exchangeabilities are discussed and furthermore here I investigate inductive supervised classifiers based on both types of exchangeabilities. I further demonstrate that the classifiers based on de Finetti type exchangeability can optimally handle test items independently of each other in the presence of infinite amounts of training data while on the other hand, classifiers based on partition exchangeability still continue to benefit from joint labelling of all the test items. Additionally, it is shown that the inductive learning process for the simultaneous classifier saturates when the amount of test data tends to infinity.

This paper proposes a new methodology for performing Bayesian inference in imaging inverse problems where the prior knowledge is available in the form of training data. Following the manifold hypothesis and adopting a generative modelling approach, we construct a data-driven prior that is supported on a sub-manifold of the ambient space, which we can learn from the training data by using a variational autoencoder or a generative adversarial network. We establish the existence and well-posedness of the associated posterior distribution and posterior moments under easily verifiable conditions, providing a rigorous underpinning for Bayesian estimators and uncertainty quantification analyses. Bayesian computation is performed by using a parallel tempered version of the preconditioned Crank-Nicolson algorithm on the manifold, which is shown to be ergodic and robust to the non-convex nature of these data-driven models. In addition to point estimators and uncertainty quantification analyses, we derive a model misspecification test to automatically detect situations where the data-driven prior is unreliable, and explain how to identify the dimension of the latent space directly from the training data. The proposed approach is illustrated with a range of experiments with the MNIST dataset, where it outperforms alternative image reconstruction approaches from the state of the art. A model accuracy analysis suggests that the Bayesian probabilities reported by the data-driven models are also remarkably accurate under a frequentist definition of probability.

Mechanistic models with differential equations are a key component of scientific applications of machine learning. Inference in such models is usually computationally demanding, because it involves repeatedly solving the differential equation. The main problem here is that the numerical solver is hard to combine with standard inference techniques. Recent work in probabilistic numerics has developed a new class of solvers for ordinary differential equations (ODEs) that phrase the solution process directly in terms of Bayesian filtering. We here show that this allows such methods to be combined very directly, with conceptual and numerical ease, with latent force models in the ODE itself. It then becomes possible to perform approximate Bayesian inference on the latent force as well as the ODE solution in a single, linear complexity pass of an extended Kalman filter / smoother - that is, at the cost of computing a single ODE solution. We demonstrate the expressiveness and performance of the algorithm by training a non-parametric SIRD model on data from the COVID-19 outbreak.